Theme Essential Question: How are functions used to describe relationships that happen in our lives?
Essential Questions: How do you graph a linear function?
Standards Understand the connections between proportional relationships, lines, and linear equations
8. EE.5Graph proportional relationships, interpreting the unit rate as the slope* of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8. F.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. *
*The standards in red are the focus for this week.
Objectives
The student will use real-world problems to explore constant rate of change by creating and using tables and graphs.
The student will use real-world problems to write equations, develop tables, and graph the ordered pairs from the tables.
The student will use tables and graphs to determine if equations are linear.
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team The lessons in Unit 3 will utilize all the mathematical practices. Keep encouraging students to think mathematically according to the practices.
In this lesson, students are introduced to y = mx + b. A detailed discussion of slope and writing equations will follow in an upcoming lesson.
In the elementary grades, students explore number and shape patterns (sequences), and they use rules for finding the next term in the sequence. At this point, students describe sequences both by rules relating one term to the next and also by rules for finding the nth term directly. (In high school, students will call these recursive and explicit formulas.) Students express rules in both words and in symbols. Instruction should focus on additive and multiplicative sequences as well as sequences of square and cubic numbers, considered as areas and volumes of cubes, respectively.
When plotting points and drawing graphs, students should develop the habit of determining based upon the context, whether it is reasonable to “connect the dots” on the graph. In some contexts, the inputs are discrete, and connecting the dots can be misleading. For example, if a function is used to model the height of a stack of n paper cups, it does not make sense to have 2.3 cups, and thus there will be no ordered pairs between n = 2 and n = 3.
Students can compute the area and perimeter of different-size squares and identify that one relationship is linear while the other is not by either looking at a table of value or a graph in which the side length is the independent variable (input) and the area or perimeter is the dependent variable (output).
Extending the understanding of ratios and proportions:
Unit rates have been explored in Grade 6 as the comparison of two different quantities with the second unit - a unit of one, (unit rate). In Grade 7 unit rates were expanded to complex fractions and percents through solving multistep problems such as: discounts, interest, taxes, tips, and percent of increase or decrease. Proportional relationships were applied in scale drawings, and students should have developed an informal understanding that the steepness of the graph is the slope or unit rate.
In Grade 8, unit rates are addressed formally in graphical representations, algebraic equations, and geometry through similar triangles.
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose of illustrating how the rates of two objects can be represented, analyzed and described in different ways: graphically and algebraically. Emphasize the creation of representative graphs and the meaning of various points. Then compare the same information when represented in an equation.
Assessment Product
Students continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel.
Entry #2 (a) Determine whether the equation is linear
The sheet rock crew charges $20 per hour with a staging cost of $50.
Create a table for the sheet rock crew working: 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.
What are the independent and dependent variables?
Prepare a graph representing the number of hours and charges.
Is this a linear function? Why?
Write an equation. Use the following hint:
Total Charge = $20 times the number of hours plus $50 staging cost =+ _
Entry #2 (b) Determine whether the equation is linear
The project site is on a mountain near a river. It has been noted that every month the amount of dirt sliding into the river doubles from the month before. The first month was 1 cubic inch, the second month 2 cubic inches, and the third month 4 cubic inches.
Create a table for six consecutive months.
What is the domain and range?
Prepare a graph representing the number of months and the cubic inches of dirt sliding into the river.
Is this a linear function? Why?
Key Questions
How do you know if the linear function has a proportional relationship?
How does y = mx + b define a linear function and graph a straight line?
What are some examples of non-linear functions?
Explain how the values from the table interconnect and produce a corresponding graph.
How is the rule (equation) related to the table and graph?
3. Observable Student Behaviors
Students can distinguish between linear and nonlinear functions.
Students can create tables and graphs from a given word problem.
Student will be familiar with the slope-intercept form of a linear equation,
y = mx + b.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Constant rate of change, variable rate of change, linear equation, linear function
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8
Unit 2-2 (TG p.35, Student p.35-38),
ABC Mastering the Common Core in Mathematics Chapter 5-1 through 5-3, p.47-50
Gizmo Lessons
Direct Variation
Adjust the constant of variation and explore how the graph of the direct variation function changes in response.
Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)
Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.EE.5 p. 169
8.F.3 p. 189
JBHM Grade 8 GP2, Unit 4, p. 265-296
Glencoe Pre Algebra Chapter 8, Lesson 5, p. 393-397
Page 392: Algebra activity (slope and rate of change)
Glencoe Algebra I, Chapter 5, Lesson 1-2, p. 256-270
Grade: 8 Unit: 3 Week:2 Dates: 12/3-12/7
Content: Graphing Linear functions
Theme Essential Question:
How are functions used to describe relationships that happen in our lives?
Essential Questions:
How do you graph a linear function?
Standards
Understand the connections between proportional relationships, lines, and linear
equations
*The standards in red are the focus for this week.
Objectives
Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team
The lessons in Unit 3 will utilize all the mathematical practices. Keep encouraging students to think mathematically according to the practices.
In this lesson, students are introduced to y = mx + b. A detailed discussion of slope and writing equations will follow in an upcoming lesson.
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 )
In the elementary grades, students explore number and shape patterns (sequences), and they use rules for finding the next term in the sequence. At this point, students describe sequences both by rules relating one term to the next and also by rules for finding the nth term directly. (In high school, students will call these recursive and explicit formulas.) Students express rules in both words and in symbols. Instruction should focus on additive and multiplicative sequences as well as sequences of square and cubic numbers, considered as areas and volumes of cubes, respectively.
When plotting points and drawing graphs, students should develop the habit of determining based upon the context, whether it is reasonable to “connect the dots” on the graph. In some contexts, the inputs are discrete, and connecting the dots can be misleading. For example, if a function is used to model the height of a stack of n paper cups, it does not make sense to have 2.3 cups, and thus there will be no ordered pairs between n = 2 and n = 3.
Students can compute the area and perimeter of different-size squares and identify that one relationship is linear while the other is not by either looking at a table of value or a graph in which the side length is the independent variable (input) and the area or perimeter is the dependent variable (output).
Extending the understanding of ratios and proportions:
Unit rates have been explored in Grade 6 as the comparison of two different quantities with the second unit - a unit of one, (unit rate). In Grade 7 unit rates were expanded to complex fractions and percents through solving multistep problems such as: discounts, interest, taxes, tips, and percent of increase or decrease. Proportional relationships were applied in scale drawings, and students should have developed an informal understanding that the steepness of the graph is the slope or unit rate.
In Grade 8, unit rates are addressed formally in graphical representations, algebraic equations, and geometry through similar triangles.
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose of illustrating how the rates of two objects can be represented, analyzed and described in different ways: graphically and algebraically. Emphasize the creation of representative graphs and the meaning of various points. Then compare the same information when represented in an equation.
Assessment
Product
- Students continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel.
Entry #2 (a) Determine whether the equation is linear- The sheet rock crew charges $20 per hour with a staging cost of $50.
- Create a table for the sheet rock crew working: 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.
- What are the independent and dependent variables?
- Prepare a graph representing the number of hours and charges.
- Is this a linear function? Why?
- Write an equation. Use the following hint:
Total Charge = $20 times the number of hours plus $50 staging cost= + _
Entry #2 (b) Determine whether the equation is linear
Key Questions
3. Observable Student Behaviors
- Students can distinguish between linear and nonlinear functions.
- Students can create tables and graphs from a given word problem.
- Student will be familiar with the slope-intercept form of a linear equation,
y = mx + b.1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Constant rate of change, variable rate of change, linear equation, linear function
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8
- Unit 2-2 (TG p.35, Student p.35-38),
- ABC Mastering the Common Core in Mathematics Chapter 5-1 through 5-3, p.47-50
- Gizmo Lessons
- Direct Variation
Adjust the constant of variation and explore how the graph of the direct variation function changes in response.- Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.- Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)- Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.- Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)- Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
- 8.EE.5 p. 169
- 8.F.3 p. 189
- JBHM Grade 8 GP2, Unit 4, p. 265-296
- Glencoe Pre Algebra Chapter 8, Lesson 5, p. 393-397
- Page 392: Algebra activity (slope and rate of change)
- Glencoe Algebra I, Chapter 5, Lesson 1-2, p. 256-270
Highly Recommended:http://illustrativemathematics.org/illustrations/129 (EE.5)
http://illustrativemathematics.org/illustrations/57 (EE.5)
http://illustrativemathematics.org/illustrations/55 (EE.5)
http://illustrativemathematics.org/illustrations/86 (EE.5)
http://illustrativemathematics.org/illustrations/184 (EE.5)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Homework
Suggested:
Terminology for Teachers
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources
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SMART Board Lessons, Promethean Lessons
Websites
Other Activities, etc.
Language
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Matrix
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Home K-2
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Unit 2
Unit 3
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